# A Simple Proof Of Bell's Theorem

by alt
Tags: bell, proof, simple, theorem
P: 1,414
 Quote by ThomasT So, you think that most physicists would say that nature is not evolving according to local and discoverable underlying dynamics?[
 Quote by JesseM Yes, virtually all of them would say this.

 Quote by JesseM Why do you have an 0.5 in front of cos^2 (a-b)?
Why are you even asking this? This is what I mean by obfuscation. Obviously, if you have a source emitting randomly polarized photons, then if you place a polarizer between the emitter and the detector, then the coeffiicient of transmission with the polarizer is 1/2 the photon flux without the polarizer.

 Quote by JesseM I'm not sure of the exact "original setup" you're thinking of ...
It was clear enough.

 Quote by JesseM ... but if the photons are entangled in such a way that identical measurements on each will yield the same results with probability 1, then if both polarizers are at the same angle so a-b=0, then the probability of getting the same result (either both passing through the polarizers, or both being reflected) should be cos^2(0) = 1, not 0.5 cos^2(0) = 0.5.
Don't be ridiculous. And unnecessarily obfuscating. The addition of a single polarizer between the emitter and the detector cuts the photon flux by 1/2.

 Quote by JesseM Assuming you should have written cos^2 (a-b) for the original setup, your altered setup above will not give the same results.
The prediction for the original setup will be .5cos^2|a-b| in the ideal. The altered setup, the polariscopic setup, will give the same results.

 Quote by ThomasT But Herbert, vis Bell, requires that the polarimeter produce a linear relationship between |a-b| and rate of detection.
 Quote by ThomasT This is why I said that light was being required by Herbert, vis Bell, to behave in a way contrary to what thousands of polarimetric experiments have shown.
Here's what you've failed to address so far. Bell and Herbert say that the number of mismatches at 30 degrees (a-b) plus the number of mismatches at 30 degrees should be less than or equal to the number of mismatches at 60 degrees. But, according to Malus Law the number of mismatches at 60 degrees should be greater than the number of mismatches at 30 degrees + the number of mismatches at 30 degrees. So, what you and Bell and Herbert are saying is contrary to 200 years of applied optics.

Of course, there's the possibility that OPTICAL Bell tests have nothing to do with OPTICS (per DrChinese et al.), but I think that that's just clutching at straws.

 Quote by JesseM If there is some specific thing you have a question about we can discuss it, but I don't really feel like going through a detailed analysis of everything in those papers.
Nor do I. In fact, I'm quite saturated with this topic for now and am quite amenable to letting it go for a while. I thank you for your detailed responses. And perhaps we will discuss this in future threads.
PF Gold
P: 5,299
 Quote by Loren Booda Paraphrased from Nick Herbert's book Quantum Reality, the following (borrowed from the OP's http://quantumtantra.com/bell2.html) is the best (simplest) account I have found to date for the Bell inequality (SPOT stands for Single Photon Orientation Tester): ... Just beautiful.
Thanks for sharing that! You are right, it is simple and beautiful!!
P: 8,470
Did you try the experiment I suggested of searching for "bell's theorem" plus "inequality" on google books? Like I said before, "would you disagree that if physics textbooks all present some theoretical conclusion as a fact, then it's a safe bet that this conclusion is one that the majority of physicists would agree with?"

And again, do a little self-reflection, do you really believe that as an amateur who has made no formal study of physics, you really have a rational basis for being confident that I (and DrChinese and RUTA and others) are wrong about the typical beliefs of physicists? You have cited only two papers with significant overlap in authorship, do you think that's enough to justify the conclusion that a significant fraction of the physics community disagrees with Bell's conclusions? Given how strongly you seem to want Bell's theorem to be wrong, do you think you are immune to effects like confirmation bias, not to mention the Dunning-Kruger effect?
 Quote by ThomasT Why are you even asking this? This is what I mean by obfuscation.
Please stop leaping to such mean-spirited conclusions, my questions are completely genuine.
 Quote by ThomasT Obviously, if you have a source emitting randomly polarized photons, then if you place a polarizer between the emitter and the detector, then the coeffiicient of transmission with the polarizer is 1/2 the photon flux without the polarizer.
To help explain my confusion, note that we aren't talking about the photon flux at an individual source, we're talking about the coincidence rate for two detectors at different locations A and B--you said "And the coincidence rate, ie. F(AB), is the same as with the original setup, .5 cos^2 |a-b|, in the ideal." In the "original setup" of the Bell test I was familiar with, when the polarizers were set to the same angle, then if there was a detection at A there'd be a probability of 1 that you'd also see a detection at B. I thought that would mean a "coincidence rate" of 1 as well. However, in the past I've normally only looked at the theoretical side of Bell's theorem and not the experimental side, so it seems like I've actually misunderstood what the "coincidence rate" equations are giving us. Looking at this paper, the author writes on p. 3:
 R(a,b) is the rate of coincidences with polarizer I in orientation a and polarizer II in orientation b ... and R0 is the coincidence rate with the two polarizers removed.
From a theoretical perspective it may be natural to think of coincidences just in terms of the probability that both members of the pair will respond the same way to the polarizers (which is how I was thinking of it, and some explanations of Bell's theorem do involve this sort of probability), but if they give R0 as the "coincidence rate with the two polarizers removed" that's presumably not what they're talking about since with no polarizers to block out the photons, efficient detectors would always detect both members of the pair with probability 1. Instead it seems that they are actually talking about a "rate" at which photons are being detected in time, like 6 photons/second or something. So in this case, the "coincidence rate" with polarizers at the same angle would naturally be half of R0, in spite of the fact that anytime an entangled photon makes it through polarizer A, its entangled twin is guaranteed to make it through polarizer B with probability 1. So if we want to know the coincidence rate expressed as a fraction of R0, the rate with no polarizers in place, it would indeed be 0.5 cos^2 (a-b).

So, sorry about the mistake, but there was really no need to accuse me of obfuscation since my confusion was genuine. And in my defense, I don't think there is perfect uniformity in how different authors use the phrase "coincidence rate", for example this page says:
 Suppose, for example, that we misalign the angles of the two polarization filters so that the angle between the polarization directions of the two filters is a=a1-a2. We measure the coincidence rate R(a), as compared to the rate Ro when the filters are perfectly aligned. ... Quantum mechanics, on the other hand, predicts that the coincidence rate R(a1,a2) depends only on the relative angle a=a1-a2 between the two polarization directions, and that R(a) obeys Malus' Law. In other words, quantum mechanics predicts that R(a1,a2)=R(a)=RoCos2(a).
So here the equation does not include any term giving the coincidence rate when the filters are removed, in front of cos^2 it puts Ro, the rate when they are in place at the same angle. And presumably this would be a literal rate of photons/second, not a dimensionless 0.5.

Anyway, with the understanding that we're looking at the coincidence rate with polarizers at angles a and b expressed as a fraction of the rate of detection with no polarizers in place, I agree that your setup--where photons going towards A meet no polarizers, while photons going towards B encounter two polarizers at angles a and b in succession--will indeed yield the same coincidence rate as with the normal setup where the photon going towards B encounters a polarizer at angle b while the photon going towards A encounters a polarizer at angle a. However, the crucial point is that your setup does not meet the experimental conditions specified by Bell, which include the fact that there must be a spacelike separation between the random choice of angle a for the polarizer that photon #1 encounters and the event of photon #2 encountering the other polarizer at angle b. This spacelike separation is crucial, because in a local universe it means there can be no causal influence between the choice of angle a and the event of the photon either passing through or being reflected by b. This is why it would be quite possible to reproduce the 0.5 cos^2 (a-b) relationship you describe using classical optics (you'd just need a detector that gave a binary yes/no result depending on whether the light that reached it was above a certain threshold), but it would be impossible to reproduce the 0.5 cos^2 (a-b) relationship in a classical optics experiment that actually satisfied Bell's experimental conditions, including the part about the spacelike separation.

And this is why the following argument is just a giant strawman:
 Quote by ThomasT Here's what you've failed to address so far. Bell and Herbert say that the number of mismatches at 30 degrees (a-b) plus the number of mismatches at 30 degrees should be less than or equal to the number of mismatches at 60 degrees.
Yes, for the specific experimental setup described under the assumption of local realism, not for any arbitrary experimental setup that doesn't match the one they assumed.
 Quote by ThomasT But, according to Malus Law the number of mismatches at 60 degrees should be greater than the number of mismatches at 30 degrees + the number of mismatches at 30 degrees.
No, the classical Malus' law does not predict this, not in the setup that Bell and Herbert described. The fact that you can find a completely different setup where the classical Malus' law does predict mismatches at 60 is greater than the sum of mismatches at 30 is just a strawman argument, since Bell never argued that his inequalities should apply in any experiments other than ones meeting the conditions he specified. Since classical electromagnetism is a local realist theory, it would indeed be impossible to replicate the 0.5 cos^2 (a-b) in a classical optics experiment that actually matched Bell's setup.
 Sci Advisor PF Gold P: 5,299 OK, how about this one: 1. Take 2 classical sets of binary observables that are uncorrelated. Say, 2 stacks of 100 coins which were randomly flipped to H or T and then paired. One stack is Alice, the other is Bob. About 50 pairs will match (ideal case), about 50 will mismatch, so 0 net correlation. 2. Find a 3rd set which has this property: it is equally correlated to Alice and Bob (in the ideal case, ignore if off by 1). There should be such a set. 3. What is the correlation of that set to Alice and Bob? Classically, it MUST be exactly .5 (within some confidence level related to sample size). Because the midpoint of 100% matches and 50% matches is 75% matches, which is a correlation of 50% (75% matches less 25% mismatches). 4. However, in the quantum version of this (PDC Type I entangled photon pairs), the correlation is more like .7 (85% matches less 15% mismatches). Imagine Alice and Bob at 0 and 45 degrees. They will not be correlated at all. But their midpoint, 22.5 degrees, is 70% correlated (85% matched) to Alice AND 70% correlated (85% matched) to Bob. That's impossible with any classical set; ergo there is no classical set. Or look at it from the other perspective: 1. I have 100 tossed coins (data values for a set at 22.5 degrees). Is it possible to obtain 2 different sets by turning over 15 coins from each set so that those new sets (Alice and Bob) are maximally different? Certainly, but how different can they be? Classically and quantum mechancially, you get different answers! 2. Classically, you will have sets that are no more than 30% different (15 changed in Alice + 15 changed in Bob out of 100 original). 3. In the QM world: When Alice is 0 degrees and Bob is 45 degrees (the farthest points where they are 15% different than the starting point of 22.5 degrees), well, those points are completely uncorrelated with each other (i.e. 50% different). Ergo, there are no classical datasets which have this simple observed property: 15%+15%=50%.
P: 1,414
 Quote by DrChinese OK, how about this one: 1. Take 2 classical sets of binary observables that are uncorrelated. Say, 2 stacks of 100 coins which were randomly flipped to H or T and then paired. One stack is Alice, the other is Bob. About 50 pairs will match (ideal case), about 50 will mismatch, so 0 net correlation. 2. Find a 3rd set which has this property: it is equally correlated to Alice and Bob (in the ideal case, ignore if off by 1). There should be such a set. 3. What is the correlation of that set to Alice and Bob? Classically, it MUST be exactly .5 (within some confidence level related to sample size). Because the midpoint of 100% matches and 50% matches is 75% matches, which is a correlation of 50% (75% matches less 25% mismatches). 4. However, in the quantum version of this (PDC Type I entangled photon pairs), the correlation is more like .7 (85% matches less 15% mismatches). Imagine Alice and Bob at 0 and 45 degrees. They will not be correlated at all. But their midpoint, 22.5 degrees, is 70% correlated (85% matched) to Alice AND 70% correlated (85% matched) to Bob. That's impossible with any classical set; ergo there is no classical set. Or look at it from the other perspective: 1. I have 100 tossed coins (data values for a set at 22.5 degrees). Is it possible to obtain 2 different sets by turning over 15 coins from each set so that those new sets (Alice and Bob) are maximally different? Certainly, but how different can they be? Classically and quantum mechancially, you get different answers! 2. Classically, you will have sets that are no more than 30% different (15 changed in Alice + 15 changed in Bob out of 100 original). 3. In the QM world: When Alice is 0 degrees and Bob is 45 degrees (the farthest points where they are 15% different than the starting point of 22.5 degrees), well, those points are completely uncorrelated with each other (i.e. 50% different). Ergo, there are no classical datasets which have this simple observed property: 15%+15%=50%.
There's a difference in experiments involving coins and experiments involving light. Does that difference imply nonlocality? I don't think it does.
PF Gold
P: 1,662
 Quote by Loren Booda Paraphrased from Nick Herbert's book Quantum Reality, the following (borrowed from the OP's http://quantumtantra.com/bell2.html) is the best (simplest) account I have found to date for the Bell inequality (SPOT stands for Single Photon Orientation Tester): ... Just beautiful.
Yes, it’s very beautiful.

I don’t wanna be a "party pooper"... but somehow I wonder if this is really the whole truth...
 [Nick Herbert] What happened was that I ended up producing one of the simplest proofs of Nature's necessary deep non-locality.
...because John Bell himself used exactly the same example in a lecture in 1990...
 N(+30°, -30°) ≤ N(+30°, 0°) + N(0°, -30°)
And this version is even shorter!
PF Gold
P: 1,662
 Quote by ThomasT Wrt what most physicists would say, it's an open empirical question. My own sampling of the physics community has lead me to believe that most physicists would say that nature is evolving according to underlying dynamical principles in accordance with the principle of local action. But of course it wasn't a large sample, and the responses were maybe not representative of the responses one might get from thousands of physicists across all the sub-fields in physics. Some specialties might be more predisposed to liking certain ideas or thinking in certain terms (eg., that nature is local -- or nonlocal) than others.
I don’t think your 'sample' is representative:
 Quote by RUTA When I first entered the foundations community (1994), there were still a few conference presentations arguing that the statistical and/or experimental analyses of EPR-Bell experiments were flawed. Such talks have gone the way of the dinosaurs. Virtually everyone agrees that the EPR-Bell experiments and QM are legit, so we need a significant change in our worldview. There is a proper subset who believe this change will be related to the unification of QM and GR :-)
RUTA is a working PhD Professor of Physics.

 Quote by ThomasT I'm going to resist the temptation to reply to these most recent Bell threads for as long as it takes for me to catch up on my reading. ... Maybe nature is nonlocal
ThomasT, I think it’s just great that you have found the word "Maybe"! I’m not a scientist, but I would guess that the M-word is the most useful of all in science!!

Happy reading, and remember – no one is saying that it’s proved beyond any doubt that the world is nonlocal. We are just saying that Local Realism (LR) cannot reproduce all the predictions of QM and the results of all performed EPR-Bell experiments so far.

A little 'advice' on subjects to penetrate, while you’re reading, is the correlation on the relative angle between Alice & Bob:

Angle	Alice	Bob
------------------------
0º	100%	100%
22.5º	85%	85%
45º	50%	50%
67.5º	15%	15%
90º	0%	0%
• And don’t forget: Alice & Bob is outside each others lightcone when the final relative angle is set.

• And don’t forget: We will not get this statistics on not entangled photons.

• And don’t forget: This statistics would also be a fact if someone will be able to do EPR-Bell experiments with electrons instead of photons – Bell's Theorem is not restricted to optics.
Take care!
P: 1,414
I have to agree. It was a sample of professional physicists, abeit none in the foundations field. So, pretty much not familiar with Bell stuff, and, as experimentalists, probably biased toward a 'local' worldview.

 Quote by RUTA When I first entered the foundations community (1994), there were still a few conference presentations arguing that the statistical and/or experimental analyses of EPR-Bell experiments were flawed. Such talks have gone the way of the dinosaurs. Virtually everyone agrees that the EPR-Bell experiments and QM are legit, so we need a significant change in our worldview. There is a proper subset who believe this change will be related to the unification of QM and GR :-)
 Quote by DevilsAvocado RUTA is a working PhD Professor of Physics.
The argument isn't that the experiments or the data analyses are flawed. So, I agree with RUTA on this. The argument is that the logic involved in assuming that violations of BIs allow inferences about fundamental reality is flawed. And so I don't agree with RUTA that the results of Bell tests require a significant change in our worldview.

 Quote by DevilsAvocado ThomasT, I think it’s just great that you have found the word "Maybe"! I’m not a scientist, but I would guess that the M-word is the most useful of all in science!!
You don't have to be a professional scientist to be a scientist. I think you have a scientific mind. It's all in how you approach learning the truth of things. And, I agree that the willingness to say "maybe", as well as "I don't know" and "I don't understand" are very important prerequisites to learning.

But I like ruffling a few feathers now and then.

 Quote by DevilsAvocado Happy reading, and remember – no one is saying that it’s proved beyond any doubt that the world is nonlocal. We are just saying that Local Realism (LR) cannot reproduce all the predictions of QM and the results of all performed EPR-Bell experiments so far.
Well, then we pretty much agree.

 Quote by DevilsAvocado A little 'advice' on subjects to penetrate, while you’re reading, is the correlation on the relative angle between Alice & Bob: Code: Angle Alice Bob ------------------------ 0º 100% 100% 22.5º 85% 85% 45º 50% 50% 67.5º 15% 15% 90º 0% 0% * And don’t forget: Alice & Bob is outside each others lightcone when the final relative angle is set.
The loophole stuff is a red herring. The experiments are fine. It's all about the logic. The usual interpretation is a false dichotomy. But, that's just my current opinion, and might well be wrong. (I'm feeling very intellectually humble today for some reason. There will be no swindlin' today. Of course, tomorrow's another day.)

 Quote by DevilsAvocado * And don’t forget: We will not get this statistics on not entangled photons.
Of course. That's how the presence of entanglement is inferred -- from the statistical results.

 Quote by DevilsAvocado * And don’t forget: This statistics would also be a fact if someone will be able to do EPR-Bell experiments with electrons instead of photons – Bell's Theorem is not restricted to optics.
I'll definitely keep that in mind.

And this really is my last post on this stuff for a while (I didn't feel like reading this morning).
PF Gold
P: 1,662
 Quote by ThomasT I have to agree.
Great!
 Quote by ThomasT So, I agree with RUTA on this.
Fantastic!
 Quote by ThomasT I think you have a scientific mind.
Thanks very much!
 Quote by ThomasT But I like ruffling a few feathers now and then.
This is very reassuring... I thought you were ill there for a while...? (joke! )

(... I’m sharpening my teeth ... )
 Quote by ThomasT Well, then we pretty much agree.
Cool!
 Quote by ThomasT The loophole stuff is a red herring.
Agree.
 Quote by ThomasT I'll definitely keep that in mind.
Very good, it will bring benefits.
 Quote by ThomasT And this really is my last post on this stuff for a while
See ya later, Terminator!
PF Gold
P: 5,299
 Quote by ThomasT Of course. That's how the presence of entanglement is inferred -- from the statistical results.
Entangled particles exist in a superposition of states. Classical particles do not exist in a superposition of states, they are in a single state. Classical of course meaning local realistic. So it seems pretty straightforward to me that finding particle pairs that have superposition statistics (i.e. Bell states) is an absolute counterexample to classical ideas. If, in fact, the particles were in mixed (i.e. classical) states, then you would get different statistics (Product state).

Again, the point is that entanglement (as a state) is incompatible with local realism.
 PF Gold P: 1,662 DrC, I’m looking for the stuff on "remote entanglement" (particles that never was in contact), but I can’t find it on PF?
PF Gold
P: 5,299
 Quote by DevilsAvocado DrC, I’m looking for the stuff on "remote entanglement" (particles that never was in contact), but I can’t find it on PF?
There are a batch of related ones from some of the top teams. Search on entanglement swapping, but here are a few:

http://arxiv.org/abs/quant-ph/0609135

We report for the first time in an ancilla-free process a non-local entanglement between two single photons which do not meet. For our experiment we derive a simple and efficient method to entangle two single photons using post-selection technology. The photons are guided into an interferometer setup without the need for ancilla photons for projection into the Bell-states. After passing the output ports, the photons are analyzed using a bell state analyzer on each side. The experimental data clearly shows a non-local interaction between these photons, surpassing the limit set by the CHSH-inequality with an S-value of 2.54 and 24 standard deviations.

http://arxiv.org/abs/0809.3991

Entanglement swapping allows to establish entanglement between independent particles that never interacted nor share any common past. This feature makes it an integral constituent of quantum repeaters. Here, we demonstrate entanglement swapping with time-synchronized independent sources with a fidelity high enough to violate a Clauser-Horne-Shimony-Holt inequality by more than four standard deviations. The fact that both entangled pairs are created by fully independent, only electronically connected sources ensures that this technique is suitable for future long-distance quantum communication experiments as well as for novel tests on the foundations of quantum physics.

http://arxiv.org/abs/quant-ph/0409093

We report the first experimental realization of entanglement swapping over large distances in optical fibers. Two photons separated by more than two km of optical fibers are entangled, although they never directly interacted. We use two pairs of time-bin entangled qubits created in spatially separated sources and carried by photons at telecommunication wavelengths. A partial Bell state measurement is performed with one photon from each pair which projects the two remaining photons, formerly independent onto an entangled state. A visibility high enough to violate a Bell inequality is reported, after both photons have each travelled through 1.1 km of optical fiber.

http://arxiv.org/abs/0911.1314

Quantum systems that have never interacted can become nonlocally correlated through a process called entanglement swapping. To characterize nonlocality in this context, we introduce local models where quantum systems that are initially uncorrelated are described by uncorrelated local variables. While a pair of maximally entangled qubits prepared in the usual way (i.e., emitted from a common source) requires a visibility close to 70% to violate a Bell inequality, we show that an entangled pair generated through entanglement swapping will already violate a Bell inequality for visibilities as low as 50% under our assumption.

---------------------------------

Although these experiments don't show it, you can entangle photons after they are detected... and you can even entangled photons that never existed at the same time.
PF Gold
P: 1,662
 Quote by DrChinese There are a batch of related ones from some of the top teams. Search on entanglement swapping, but here are a few:
BIG Thanks!

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